On the number of touching pairs in a set of planar curves

2015/11/17 by Györgyi, Péter, Hujter, Bálint, Kisfaludi-Bak, Sándor
#52C30 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1511.05425

Abstract

Given a set of planar curves (Jordan arcs), each pair of which meets -- either crosses or touches -- exactly once, we establish an upper bound on the number of touchings. We show that such a curve family has O(t²n) touchings, where t is the number of faces in the curve arrangement that contains at least one endpoint of one of the curves. Our method relies on finding special subsets of curves called quasi-grids in curve families; this gives some structural insight into curve families with a high number of touchings.

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